#148492

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2073
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6)(7,19)(8,25,30,59)(9,32)(10,15)(11,37)(12,18)(13,21)(14,28,46,64)(16,35)(17,34)(20,39,54,67)(22,41)(23,57)(24,49)(26,60,52,66)(27,62,29,47)(31,45,63,44)(33,50)(36,48)(38,40)(42,56)(43,65,51,53)(55,58,61,68);(1,2)(3,4)(5,7)(6,13)(8,26)(9,33)(10,21)(11,38)(12,24)(14,43)(15,40)(16,49)(17,50)(18,42)(19,32)(20,29)(22,35)(23,34)(25,58)(27,54)(28,55)(30,52)(31,65)(36,56)(37,41)(39,45)(44,67)(46,51)(47,66)(48,57)(53,63)(59,68)(60,62)(61,64);(2,4)(5,8,9,27,34,63,36,64,12,30,35,29,11,31,10,28)(6,14,15,44,37,62,16,25,18,46,48,45,17,47,32,59)(7,20,21,52,38,61,22,53,24,54,56,26,23,55,33,65)(13,39,19,51,50,58,57,60,42,67,49,43,41,68,40,66)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,h^b=hi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55688);